3.910 \(\int \frac{x^6}{\left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{x^3}{2 \sqrt{1-x^4}}+\frac{3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

x^3/(2*Sqrt[1 - x^4]) - (3*EllipticE[ArcSin[x], -1])/2 + (3*EllipticF[ArcSin[x],
-1])/2

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Rubi [A]  time = 0.0606169, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^3}{2 \sqrt{1-x^4}}+\frac{3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^6/(1 - x^4)^(3/2),x]

[Out]

x^3/(2*Sqrt[1 - x^4]) - (3*EllipticE[ArcSin[x], -1])/2 + (3*EllipticF[ArcSin[x],
-1])/2

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Rubi in Sympy [A]  time = 11.0147, size = 32, normalized size = 0.91 \[ \frac{x^{3}}{2 \sqrt{- x^{4} + 1}} - \frac{3 E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{2} + \frac{3 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-x**4+1)**(3/2),x)

[Out]

x**3/(2*sqrt(-x**4 + 1)) - 3*elliptic_e(asin(x), -1)/2 + 3*elliptic_f(asin(x), -
1)/2

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Mathematica [A]  time = 0.0572107, size = 32, normalized size = 0.91 \[ \frac{1}{2} \left (\frac{x^3}{\sqrt{1-x^4}}+3 F\left (\left .\sin ^{-1}(x)\right |-1\right )-3 E\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(1 - x^4)^(3/2),x]

[Out]

(x^3/Sqrt[1 - x^4] - 3*EllipticE[ArcSin[x], -1] + 3*EllipticF[ArcSin[x], -1])/2

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Maple [A]  time = 0.014, size = 54, normalized size = 1.5 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{3\,{\it EllipticF} \left ( x,i \right ) -3\,{\it EllipticE} \left ( x,i \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-x^4+1)^(3/2),x)

[Out]

1/2*x^3/(-x^4+1)^(1/2)+3/2*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*(Elliptic
F(x,I)-EllipticE(x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^6/(-x^4 + 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x^{6}}{{\left (x^{4} - 1\right )} \sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral(-x^6/((x^4 - 1)*sqrt(-x^4 + 1)), x)

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Sympy [A]  time = 2.38722, size = 31, normalized size = 0.89 \[ \frac{x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-x**4+1)**(3/2),x)

[Out]

x**7*gamma(7/4)*hyper((3/2, 7/4), (11/4,), x**4*exp_polar(2*I*pi))/(4*gamma(11/4
))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^6/(-x^4 + 1)^(3/2), x)